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Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang
To cite this version:
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. 2010.
�hal-00496950v2�
BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF IN THE WHOLE SPACE: I, GLOBAL EXISTENCE FOR SOFT POTENTIAL
R. ALEXANDRE, Y. MORIMOTO, S. UKAI, C.-J. XU, AND T. YANG
Abstract. It is known that the singularity in the non-cutoffcross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoffBoltzmann equation for general physical cross sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.
Contents
1. Introduction 2
2. non-isotropic norm and estimates of linearized collision operators 7
2.1. Bounds on the non-isotropic norm 8
2.2. Equivalence to the linearized operator 16
2.3. Non-isotropic norms with different kinetic factors 20 3. Estimates of nonlinear collision operator in velocity space 23
3.1. Upper bounds in general case 23
3.2. A simple proof of Theorem 1.2 forγ >−3/2 29
3.3. Proof of Theorem 1.2 34
3.4. Estimation of commutators 39
4. Functional estimates in full space 41
4.1. Estimations without weight 41
4.2. Estimation with weight 43
4.3. Estimation with modified weight 44
4.4. Weighted coercivity of the linearized operator 46
5. Local existence 48
5.1. Classical solutions 49
5.2. L2-solutions 51
6. Global solutions 52
6.1. L2-solutions 52
6.2. Classical solutions 59
7. Appendix 64
References 66
2000 Mathematics Subject Classification. 35A05, 35B65, 35D10, 35H20, 76P05, 84C40.
Key words and phrases. Boltzmann equation, coercivity estimate, non-cutoffcross sections, global existence, non-isotropic norm, soft potential.
1
1. Introduction
This is the first part of a series of papers related to the inhomogeneous Boltzmann equa- tion without angular cut-off, in the whole space and for general physical cross-sections.
This global project is a natural continuation of our previous study [7] which was special- ized to Maxwellian type cross sections.
In this part, we first establish an essential coercivity estimate of the linearized collision operator, in the framework of general cross sections. As shown in [7, 8] for the special Maxwellian case, this estimate will play an important role for the related Cauchy problem.
Based on this estimation, together with Part II [9], we will prove the global existence of classical non-negative solutions to the Boltzmann equation without angular cutoff, for the soft and hard potentials respectively, so that we are able to cover a general physical setting.
Finally, in the paper [10], we will study the qualitative properties of solutions, that include full regularity, non-negativity, uniqueness and convergence rates to the equilibrium. This series of works establish a satisfactory theory on the well-posedness and full regularity of classical solutions.
In our presentation, we consider the problem in the physical case with dimension 3.
However, our results hold true for any dimension bigger than 2.
Consider
(1.1) ft+v· ∇xf =Q( f,f ), f|t=0= f0.
Here, f = f (t,x,v) is the density distribution function of particles, having position x∈R3 and velocity v ∈ R3 at time t. The right hand side of (1.1) is the Boltzmann bilinear collision operator, which is given in the classicalσ−representation by
Q(g,f )= Z
R3
Z
S2
B (v−v∗, σ)g′∗f′−g∗f dσdv∗,
where f∗′= f (t,x,v′∗),f′= f (t,x,v′),f∗= f (t,x,v∗),f = f (t,x,v), and forσ∈S2, v′=v+v∗
2 +|v−v∗|
2 σ, v′∗=v+v∗
2 −|v−v∗| 2 σ,
which gives the relation between the post and pre collisional velocities that follow from the conservation of momentum and kinetic energy.
For monatomic gas, the non-negative cross section B(z, σ) depends only on|z|and the scalar product|zz| ·σ. As in our previous works, we assume that it takes the form
(1.2) B(v−v∗,cosθ)= Φ(|v−v∗|)b(cosθ), cosθ= v−v∗
|v−v∗| · σ , 0≤θ≤ π 2, in which it contains a kinetic factor given by
Φ(|v−v∗|)= Φγ(|v−v∗|)=|v−v∗|γ, and a factor related to the collision angle containing a singularity,
b(cosθ)≈Kθ−2−2s when θ→0+, for some constant K>0.
An important example of this singular cross section is the inverse power law potential ρ−r with r>1,ρbeing the distance between two interacting particles, in which s = 1r ∈ ]0,1[ andγ=1−4s∈]−3,1[, cf. [12].
In the theory on the non-cutoffBoltzmann equation, the sign ofγ+2s plays a crucial role. Hence, from now on, the case whenγ+2s ≤ 0 is referred to the non-cutoffsoft
potential, while the case γ+2s > 0 to the non-cutoffhard potential. Note that this is different from the traditional classification on the index for the inverse power law.
In our present series of works, the well-posedness theory established applies to the general cross-sections withγ >−3 and 0<s<1, that includes the inverse power law as a special example. Note thatγ >−3 is needed for the Boltzmann operator to be well-posed, cf. [40].
Being concerned with a close to equilibrium framework, as in [7], the setting of the problem can be formulated as follows. First of all, without loss of generality, consider the perturbation around a normalized Maxwellian distribution
µ(v)=(2π)−32e−|v|
2 2 , by setting f =µ+√µg. Since Q(µ, µ)=0, we have
Q(µ+√µg, µ+√µg)=Q(µ, √µg)+Q(√µg, µ)+Q(√µg, √µg).
Denote
Γ(g,h)=µ−1/2Q(√µg, √µh).
Then the linearized Boltzmann operator takes the form
Lg=L1g+L2g=−Γ(√µ ,g)−Γ(g, √µ).
Now the original problem (1.1) is reduced to the Cauchy problem for the perturbation g (1.3)
( gt+v· ∇xg+Lg= Γ(g,g), t>0, g|t=0=g0.
This close to equilibrium framework is classical for the Boltzmann equation with an- gular cutoff, but much less is known for the Boltzmann equation without angular cutoff, though the spectrum of the linearized operator without angular cut-offwas analyzed a long time ago by Pao in [33].
However, since the late 1990s, the regularizing effect on the solution, produced by the singularity of the cross-section, has become reachable by rigorous analysis. Let us mention the systematic work on the entropy dissipation method initiated by Alexandre [1] and de- veloped firstly by Lions [26], and then by many others, cf [3, 39, 40] and references therein.
Since then, various works have been done on deriving the coercivity estimates in different settings and in different norms for different purposes. In particular, this kind of coercivity estimates has displayed some non-isotropic property in the very loose sense that, on one hand one gets a gain of the regularity in Sobolev norm of fractional order; and on the other hand, one also get a gain the moment to some fractional power in the velocity variable, cf.
[2, 3, 5, 6, 7, 16, 21, 22, 24, 31, 32, 38, 39, 40] and references therein. Furthermore, these coercivity estimates have been proven to be very useful in getting the global existence and gain of full regularity in all variables for the Boltzmann equation without angular cutoff, as shown in our previous work [7]. For details about the recent progress in some of the directions mentionned previously, readers are referred to the survey paper by Alexandre, [2].
Since the coercivity estimate plays an important role in the study on the angular non- cutoffBoltzmann equation, such estimate in terms of the indicesγand s, has been pursued by many people. One of the purposes of this paper is to present a precise estimate that gives the essential properties of this singular behavior, that will be stated in the next theorem. Let us note that this result is proved in a general setting and it improves on previous results, such as those obtained in [5, 6, 7, 31, 32]. And this estimate will be used herein and in our
papers [9, 10] on the global existence in the hard potential case, and qualitative study of solutions.
To derive the desired coercivity estimate, we generalize the non-isotropic norm intro- duced in [7] as
|||g|||2=
$
Φ(|v−v∗|)b(cosθ)µ∗ g′−g2
+
$
Φ(|v−v∗|)b(cosθ)g2∗ p
µ′ −√µ2
, where the integration is overR3
v×R3
v∗×S2
σ. Note that it is a norm with respect to the velocity variable v ∈ R3 only. We can compare this non-isotropic norm with classical weighted Sobolev norms, see precisely Proposition 2.1.
The introduction of this norm was motivated by the study on the Landau equation which can be viewed as the grazing limit of the Boltzmann equation. It is known that for the Landau equation, see for example [19], that the essential norm in order to capture the dissipation of the linearized Landau operator can be defined just as the Dirichlet form of the linearized operator. By doing so, a norm can be well defined without loss of any dissipative information in the operator and this can be done directly for the Landau equation mainly because the corresponding Landau operator is a differential operator. However, for the Boltzmann equation without angular cutoff, the collision operator is a singular integral operator so that a direct analog is not obvious or feasible. Therefore, in the first part of this paper, we analyze the properties of the non-isotropic norm and obtain the precise coercivity estimate of the linearized collision operator. At this point, let us mention the different approach undertaken by Gressman-Strain [21, 22].
We shall use the following standard weighted Sobolev space defined, for k, ℓ∈R, as Hℓk=Hkℓ(R3v)={f ∈ S′(R3v); Wℓf ∈Hk(R3v)}
and
Hℓk(R6
x,v)={f ∈ S′(R6
x,v); Wℓf ∈Hk(R6
x,v)}
where Wℓ(v)=hviℓ =(1+|v|2)ℓ/2 is always the weight for v variables. Herein, (·,·)L2 = (·,·)L2(R3v) denotes the usual scalar product in L2 = L2(R3) for v variables. Recall that L2ℓ =H0ℓ.
We shall use also in the following two different Sobolev spaces, one with x-derivatives only, another one with x,v derivatives and weight in the velocity variable v. For k∈N, ℓ∈ R, let
Hℓk(R6)=n
f ∈ S′(R6
x,v) ; kfk2Hk
ℓ(R6)= X
|α|+|β|≤N
kWℓ−|β|∂αβfk2L2(R6)<+∞o , H˜ℓk(R6)=n
f ∈ S′(R6x,v) ; kfk2H˜ℓk(R6)= X
|α|+|β|≤N
kW˜ℓ−|β|∂αβfk2L2(R6)<+∞o ,
where ˜Wℓ=(1+|v|2)|s+γ/2|ℓ/2.
We recall that the linearized operatorLhas the following null space, which is spanned by the set of collision invariants:
N =Spannõ ,v1õ ,v2õ ,v3õ ,|v|2õo , that is,
Lg,g
L2(R3v)=0 if and only if g∈ N.
Theorem 1.1. Assume that the cross-section satisfies (1.2) with 0 < s <1 andγ > −3.
Then there exist two generic constants C1,C2 >0 such that for any suitable function g C1|||(I−P)g|||2≤
Lg,g
L2 ≤C2|||g|||2, where P is the L2-orthogonal projection onto the null spaceN.
This coercivity estimate of the linearized collisional operator will prove to be crucial for the global existence of classical solutions to the Boltzmann equation. For this purpose, the analysis on the nonlinear operator is necessary, and we prove the following upper bound estimate.
Theorem 1.2. For all 0<s<1, assume thatγ >max{−3,−32−2s}. Then, one has, Γ( f,g),h
L2.n
kfkL2s+γ/2|||g|||+kgkL2s+γ/2|||f|||
+min
||f||L2||g||L2s+γ/2,||f||L2s+γ/2||g||L2 o
|||h|||, for suitable functions f,g,h .
We will then concentrate on the global existence of solutions, both weak and strong, for the non-cutoffsoft potential case in the framework of small perturbation of an equilibrium state. Even though some estimates hold for the general case and will be used in the forth- coming papers, the conditionγ+2s≤0 will be imposed in the main existence results. In the Part II [9], we will then present the global existence theory for the hard potential case, that is, the conditionγ+2s > 0 imposed. Furthermore, the qualitative behavior of the solutions, such as the uniqueness, non-negativity, regularity and convergence rate to the equilibrium will be investigated in [10]. Note that both the global existence and the qual- itative study on the solution behavior were firstly investigated in [7] for the Maxwellian molecule case where a generalized uncertainty principle obtained in [5] was used.
We begin with a local existence of classical solutions that holds true in general case.
Theorem 1.3. Assume that the cross-section satisfies (1.2) withγ+2s≤0, 0<s<1 and γ >−3. Let N ≥6 andℓ≥N. For a smallε >0, if||g0||HℓN(R6)≤ε, then there exists T >0 such that the Cauchy problem (1.3) admits a solution
g∈L∞([0,T ];HℓN(R6)).
Since we are interested in getting global existence results, the next statement deals with this issue asking only for control of x derivatives.
Theorem 1.4. Assume that the cross-section satisfies (1.2) withγ+2s≤0, 0<s<1 and γ > max{−3,−32 −2s}. Let N ≥3. For a small ε > 0, ifkg0kHN(R3x;L2(R3v)) ≤ε,then the Cauchy problem (1.3) admits a global solution
g∈L∞([0,+∞[; HN(R3
x; L2(R3
v))).
The above global existence result is in a non-weighted function space without v deriva- tives in the framework of weak solutions. On the other hand, we will prove the following global existence result on classical solutions for which the proof is more involved. Note that for the qualitative study on the solution behavior, such as the regularity as will be shown in [10], solutions in a function space with x and v derivatives together with weight in v are needed. Hence, the next theorem also serves for this purpose.
Theorem 1.5. Assume that the cross-section satisfies (1.2) withγ+2s≤0, 0<s<1 and γ >max{−3,−32 −2s}. Let N ≥6, ℓ≥N. For a smallε >0, ifkg0kH˜ℓN(R6) ≤ε,then the Cauchy problem (1.3) admits a global solution
g∈L∞([0,+∞[ ; H˜ℓN(R6)).
Let us now review some related works on this topic. First of all, the well-posedness theory for the Boltzmann equation has now been well established under the Grad’s angular cutoffassumption. Under this assumption, there exist basically three frameworks of exis- tence of global solutions. The first one was initiated by Grad [18] and firstly completed by Ukai [35, 36, 38] in the framework of weighted L∞v function space for small perturbation of an equilibrium, where the spectrum analysis was used through a bootstrap argument. An important progress on the existence theory is the introduction of the renormalized solutions for large perturbation in the framework of L1v function space by DiPerna-Lions [17, 25], where the velocity averaging lemma plays a key role. Recently, solutions in L2vframework were established by macro-micro decompositions and energy method for small perturba- tion of an equilibrium, cf [19, 20, 27, 28].
However, without Grad’s angular cutoffassumption, the established mathematical theo- ries are far less. In this direction, the spectral analysis of the linearized collisional operator was studied by Pao [33]. In 1990’s, some simplified models, such as Kac’s model and the Boltzmann equation in lower dimensions with symmetry, were successfully studied, [13, 14, 15]. In 2000’s, the mathematical theory for the spatially homogeneous Boltzmann equation was satisfactorily solved, [3, 4, 16, 24, 29, 30]. For the original Boltzmann equa- tion in physical space, in the framework of renormalized solutions, the only existing result can be found in [11] where the basic existence result is still lacking. There are some local existence results, [1, 37], see also the reviews [2, 40].
Since 2006, we have been working on the original Boltzmann equation without angular cutoff, cf. [5, 6, 7]. Based on a new generalized uncertainty principle proved in [5], we developed a new approach for the regularity study. In the framework of small perturbation of an equilibrium in the whole space, the first complete global well-posedness theory and regularity were established for the Maxwellian molecule case [7]. As a continuation of these works, we successfully solve, in this series of papers, the fundamental problems, that is, existence, uniqueness, regularity, non-negativity and convergence rates of solutions, so that a complete and satisfactory mathematical theory is now established under some minimal regularity requirement on the initial data. Through this analysis, mathematical tools and techniques from harmonic analysis are used and some new ones are introduced, such as the generalized uncertainty principle and the above non-isotropic norm. Here we would like to mention that recently by using a different method, an existence result on solutions in the torus case was obtained in [21, 22, 23].
Finally, we present the main strategy of analysis in this paper. Based on the essential co- ercivity estimate on the linearized collisional operator and the non-isotropic norm proven in a first step, what is needed in this paper is then the detailed analysis on the collisional op- erator in both unweighted and weighted spaces, where its upper bounds, commutators with differentiation in v and commutators with weights in v are given. Through this analysis, we can see the role played by the parametersγand s in the cross section. With these estimates, the energy method can be applied through the macro-micro decomposition analysis intro- duced in [19, 20]. Basically, the microscopic component of the solution is controlled by the essential coercivity estimate on the linearized collisional operator in the non-isotropic norm, while the dissipation on the macroscopic component is recovered by the system on the fluid functions through the macro-micro decomposition. Then the nonlinear terms are
essentially of higher order in the non-isotropic norm so that the energy estimate can be closed in the framework of small perturbation.
The rest of the paper is arranged as follows. In the next section, we extend the definition of the non-isotropic norm introduced in [7] and then state the main estimates in this paper.
The proof of the upper and lower bound estimates of the non-isotropic norm will be given in Section 3. In Section 4, we will prove the equivalence of the Dirichlet form of the linearized collison operator and the square of the non-isotropic norm. The equivalence of the non-isotropic norms with different kinetic factors and different weights will be shown in Section 5. An upper bound estimate on the nonlinear collision operator which is useful for the well-posedness theory for the Boltzmann equation will be given next. However, because of unnecessary restrictions on the values of the parameter γ, we shall amplify such estimations and obtain some new functional estimates in v variable only in another Section. The functional estimates in both v and x variables are given in Section 4. With these estimates, the local and global existence of both weak and classical solutions are given in the last two sections respectively.
Notations: Herein, letters f , g,· · · stand for various suitable functions, while C, c,· · · stand for various numerical constants, independent from functions f , g,· · · and which may vary from line to line. Notation A . B means that there exists a constant C such that A ≤ CB, and similarly for A & B. While A ∼ B means that there exist two generic constants C1,C2>0 such that
C1A≤B≤C2A.
2. non-isotropic norm and estimates of linearized collision operators
For later use, we will need to compare the original cross-section with the situation when its kinetic part is mollified. That is, for the functionΦ(z) appearing in the cross-section, we denote by ˜Φ(z) = (1+|z|2)γ2 its smoothed version. To show the dependence of the estimates on the mollified or non-mollified kinetic factor in the cross-session, we will use the notations QΦ˜ and QΦto denote the Boltzmann collisional operator when the kinetic part is ˜ΦandΦrespectively. In particular, Q=QΦ. This upper-script will be also used for other operators as well.
First of all, let us recall that Lg,g
L2=−
Γ(√µ ,g)+ Γ(g, √µ),g
L2≥0, and the definition of the non-isotropic norm
|||g|||2=
$
Φ(|v−v∗|)b(cosθ)µ∗ g′−g2
(2.1)
+
$
Φ(|v−v∗|)b(cosθ)g2∗ p
µ′ −√µ2
=J1+J2, where the integration is overR3
v×R3
v∗×S2
σ.
The following proposition gives a precise version of Theorem 1.1.
Proposition 2.1. Assume that the cross-section satisfies (1.2) with 0<s<1 andγ >−3.
Then there exist two generic constants C1,C2 >0 such that C1|||(I−P)g|||2≤
Lg,g
L2 ≤2 L1g,g
L2≤C2|||g|||2 for any suitable function g.
Concerning the lower and upper bounds of the non-isotropic norm we have
Proposition 2.2. Assume that the cross-section satisfies (1.2) with 0<s<1 andγ >−3.
Then there exist two generic constants C1,C2 >0 such that C1
kgk2Hγ/2s +kgk2L2
s+γ/2
≤ |||g|||2≤C2kgk2Hs+γ/2s
for any suitable function g.
¿From this estimate and Theorem 1.1, we can get the following estimate in classical weighted Sobolev spaces
C1
k(I−P)gk2Hγ/2s +k(I−P)gk2L2
s+γ/2
≤ Lg,g
L2 ≤C2kgk2Hss+γ/2 . (2.2)
In the following, we will use the lower scriptΦon the non-isotropic norm, and so use the notation|||g|||Φif we need to specify its dependence on the kinetic factorΦ. Notations J1Φ,J2Φwill be also used for the same purpose.
Part of the proof on the lower bound of the non-isotropic norm given in Proposition 2.2 is essentially due to the following equivalence relations.
Proposition 2.3. Assume that the cross-section satisfies (1.2) with 0<s<1 andγ >−3.
Then we have
|||g|||Φ∼ |||g|||Φ˜.
Concerning the dependence on the indexγinΦγ=|v−v∗|γ, we have
Proposition 2.4. Assume that the cross-section satisfies (1.2) with 0<s<1 andγ >−3.
Then for anyβ >−3, we have
|||g|||Φγ∼ |||hvi(γ−β)/2g|||Φβ.
2.1. Bounds on the non-isotropic norm. This section is devoted to the proof of Proposi- tion 2.2. We will often use the following elementary estimate stated in velocity dimension n, since it will be needed for both cases n=2 and n=3.
Lemma 2.5. Let the velocity dimension be n, n ∈ N, ρ > 0, δ ∈ Rand let µρ,δ(u) = huiδe−ρ|u|2for u∈Rn. Ifα >−n andβ∈R, then we have
(2.3) Iα,β(u)=
Z
Rn|w|αhwiβµρ,δ(w+u)dw∼ huiα+β. Proof. Since we have
huiβhu+wi−|β|≤ hwiβ≤ huiβhu+wi|β|,
it suffices to show (2.3) withβ=0, by takingµρ,δ±|β|instead ofµρ,δ. Taking into account the fact thatα >−n, this estimate is obvious when|u| ≤1. If|u| ≥1, then we have
Iα,0(u)≥4−|α|huiα Z
{|u+w|≤1/2}
µρ,δ(u+w)dw&huiα,
because|u+w| ≤1/2 implies that 4−1hui ≤ |w| ≤4hui. Noticing that 2|w| ≥ hwiif|w| ≥1, we have
Iα,0(u)≤ max
|w|≤1µρ,δ(u+w) Z
{|w|≤1}|w|αdw+2|α| Z
{|w|≥1}hwiαµρ,δ(u+w)dw .
hui|δ|e−ρ|u|2/2+huiα Z
Rnhu+wiαµρ,δ(u+w)dw .huiα.
And this completes the proof of the lemma.
Recall from (2.1) that the non-isotropic norm contains two parts, denoted by J1and J2
respectively. The estimation on each part will be given in the following subsections. We start with the estimation on J2because the analysis is easier.
Let us start with the following upper bound on J2.
Lemma 2.6. Under the same assumptions as in Theorem 1.1, we have J2:=
$
b(cosθ)Φ(|v−v∗|)g2∗ p
µ′ − √µ2dvdv∗dσ.kgk2L2
s+γ/2
.
Proof. Note that J2 ≤2
$
bΦ(|v−v∗|)g2∗
µ′1/4−µ1/42
µ′1/2+µ1/2
dvdv∗dσ .
$
b|v′−v∗|γg2∗
µ′1/4−µ1/42
µ′1/2dvdv∗dσ +
$
b|v−v∗|γg2∗
µ′1/4−µ1/42
µ1/2dvdv∗dσ
=F1+F2.
By the regular change of variables v→v′, we have F1.
"
|v′−v∗|γ Z
b(cosθ) min
|v′−v∗|2θ2,1 dσ
g2∗µ′1/2dv′dv∗ .Z Z
|v′−v∗|γ+2sµ′dv′
g2∗dv∗.kgk2L2
s+γ/2
,
where we have used Lemma 2.5 in the case n = 3 to get the last inequality. A direct estimation show thats the same bound holds true for F2. And this completes the proof of
the lemma.
Remark 2.7. Note that the above lemma holds even ifΦis replaced by ˜Φby using Lemma 2.5.
We now turn to the lower bound for J2.
Lemma 2.8. Under the assumptions (1.2) with γ >−3, there exists a constant C>0 such that
J2:=
$
b(cosθ)Φ(|v−v∗|)g2∗ p
µ′ − √µ2dvdv∗dσ≥Ckgk2L2
s+γ/2
. Proof. We will apply the argument used in [39]. By shifting to theω-representation,
v′=v− (v−v∗)·ωω v′∗=v+ (v−v∗)·ωω , ω∈S2, in view of the change of variables (v,v∗)→(v∗,v), we get,
J2 =4
$
b(cosθ) sin(θ/2)Φ(|v−v∗|)g2 p
µ′∗ − √µ∗ 2
dvdv∗dω ,
because dσ = 4 sin(θ/2)dω. Then, we use the Carleman representation. The idea of this representation is to replace the set of variables (v,v∗, ω) by the set (v,v′,v′∗). Here, v,v′∈R3and v′∗ ∈Evv′, where Evv′is the hyperplane passing through v and orthogonal to v−v′. By using the formula
dv∗dω= dv′∗dv′
|v−v′|2,
cf. page 347 of [39], and by taking the change of variables (v,v′,v′∗)→(v,v+h,v+y),
with h ∈R3 and y∈Eh, where Eh is the hyperplane orthogonal to h passing through the origin inR3, we have
J2∼ Z
R3 v
Z
R3 h
Z
y∈Eh∩{|y|≥|h|}
|y|1+2s+γ
|h|1+2s g(v)2
× p
µ(v+y) −p
µ(v+y+h)2
dvdhdy
|h|2 , because
|h|=|v′−v|=|v′∗−v|tanθ
2 =|y|tanθ
2, θ∈[0, π/2], b(cosθ) sin(θ/2)Φ(|v−v∗|)∼ |v∗−v′|1+2s+γ
|v−v′|1+2s 1{|v′∗−v|≥|v′−v|}.
We decompose v = v1+v2, where v2 is the orthogonal projection of v on Eh. Since µis invariant by rotation, we may assume v = (0,0,|v|) without loss of generality. By introducing the polar coordinates
h=(ρsinϑcosφ, ρsinϑsinφ, ρcosϑ), ϑ∈[0, π], φ∈[0,2π], ρ >0,
we obtain|v1|=|v||cosϑ|,|v1+h|=| |v|cosϑ+ρ|and|v2|=|v|sinϑ. Note that if 0< ϑ≤ π/2, then
pµ(v+y) −p
µ(v+y+h)2
=µ(v2+y) p
µ(v1) −p
µ(v1+h)2
≥µ(v2+y)µ(v1) 1−e−ρ2/42
/(2π)3/2. Therefore, we have for anyδ >0
J2≥C Z
R3v
g(v)2n Z
R3
h
pµ(v1) −p
µ(v1+h)2
|h|3+2s
× Z
y∈Eh∩{|y|≥|h|}|y|1+2s+γµ(v2+y)dy dho
dv
≥C Z
R3v
g(v)2n Z π/2
π/2−1/hvi
µ(v1)
Z δ 0
1−e−ρ2/42
ρ1+2s
× Z
y∈Eh|y|1+2s+γµ(v2+y)dy
− Z
y∈Eh∩{|y|≤ρ}|y|1+2s+γµ(v2+y)dy dρ
!
sinϑdϑo dv. Since we have
Z
y∈Eh∩{|y|≤ρ}|y|1+2s+γµ(v2+y)dy≤δ2s Z
y∈Eh|y|1+γµ(v2+y)dy, ifρ≤δ , and it follows from Lemma 2.5 in the case n=2, that
Z
y∈Eh
|y|βµ(v2+y)dy∼ hv2iβ ifβ >−2,
there exist two constants C1,C2>0 such that ifρ≤δ, we have Z
y∈Eh|y|1+2s+γµ(v2+y)dy− Z
y∈Eh∩{|y|≤ρ}|y|1+2s+γµ(v2+y)dy
≥C1hv2i1+2s+γ−C2δ2shv2i1+γ. Taking a sufficiently smallδ >0 gives
J2≥C Z
R3v
g(v)2n Z π/2
π/2−1/hvi
µ(v1)
×
Z δ 0
1−e−ρ2/42
ρ1+2s dρ
hv2i1+2s+γsinϑdϑo dv
≥Cδ
Z
R3vhvi2s+γg(v)2n Z π/2
π/2−1/hvi
e−|v|2cos2ϑhvidϑo dv
≥Cδkgk2s+γ/2.
The proof of the lemma is now completed.
Remark 2.9. In the above proof, the factor|y|γcan be replaced byhyiγ, so that Lemma 2.8 is valid even ifΦis replaced by ˜Φ =hv−v∗iγ. By the above lemma together with Lemma 2.6 and the Remark after it, we can conclude
(2.4) J2Φ∼ kgk2L2s+γ/2∼J2Φ˜ .
We now turn to the estimation of the first term of the non-isotropic norm, that is, J1. We will firstly show that the singular behavior of the cross-section when v = v∗ can be smoothed out. This point is given by the following proposition.
Proposition 2.10. Under the same assumption as in Theorem 1.1, we have
JΦ1 +kgk2L2s+γ/2∼J1eΦ+kgk2L2s+γ/2.
Remark 2.11. This proposition is nothing but Proposition 2.3 by Remark 2.9.
Proof. By using similar arguments as in the proof of Lemma 2.8, it follows from the Car- leman representation that
J1Φ∼ Z
R3v
Z
R3
h
Z
y∈Eh∩{|y|≥|h|}
|y|1+2s+γ
|h|1+2s µ(v) g(v+y) −g(v+y+h)2
dvdhdy
|h|2
= Z
R3v
Z
R3
h
Z
y∈Eh∩{|y|≥|h|}
|y|1+2s+γ
|h|1+2s µ(v+y) g(v) −g(v+h)2dvdhdy
|h|2 ,
where the last equality is a direct consequence of the change of variables (v+y,y)→(v,−y).
Similarly, we have J1Φe∼
Z
R3v
Z
R3
h
Z
y∈Eh∩{|y|≥|h|}
|y|1+2shyiγ
|h|1+2s µ(v+y) g(v+h) −g(v)2
dvdhdy
|h|2 .
We claim that Z
R3 v
Z
R3 h
Z
y∈Eh∩{|y|≤|h|}
|y|1+2s+γ
|h|1+2s µ(v+y) g(v+h) −g(v)2
dvdhdy
|h|2 (2.5)
.kgk2L2s+γ/2, Z
R3v
Z
R3
h
Z
y∈Eh∩{|y|≤|h|}
|y|1+2shyiγ
|h|1+2s µ(v+y) g(v+h) −g(v)2
dvdhdy
|h|2 (2.6)
.kgk2L2
s+γ/2
.
Note carefully that the integration in these estimates is performed for ”large” values of h.
Once we admit those estimates, to conclude the proof of the lemma, it suffices to show that
G(v,h)= Z
y∈Eh|y|1+2s+γµ(v+y)dy∼ Z
y∈Eh|y|1+2shyiγµ(v+y)dy=G(v,˜ h).
We decompose v=v1+v2, where v2is the orthogonal projection of v on Eh. Then we have µ(v+y)=µ(v1)µ(v2+y), whence it follows from Lemma 2.5 together with 1+2s+γ >−2 that
G(v,h)∼µ(v1)hv2i1+2s+γ∼G(v,˜ h). It remains to show (2.5) and (2.6). We write
Z
R3v
Z
R3
h
Z
y∈Eh∩{|y|≤|h|}
|y|1+2s+γ
|h|1+2s µ(v+y) g(v+h) −g(v)2dvdhdy
|h|2
= Z
R3 v
Z
{|h|≤1}
Z
y∈Eh∩{|y|≤|h|}
+ Z
R3 v
Z
{|h|≥1}
Z
y∈Eh∩{|y|≤|h|}
=A1+A2. Take a smallδ >0 such thatγ−δ >−3. Then, in view of 1+γ−δ >−2, we have
A1≤ Z
R3v
Z
{|h|≤1}
Z
y∈Eh
|y|1+γ−δ
|h|1−δ µ(v+y) g(v+h) −g(v)2
dvdhdy
|h|2
= Z
R3v
µ(v1) Z
{|h|≤1}
Z
y∈Eh|y|1+γ−δµ(v2+y)dy
g(v+h) −g(v)2 dh
|h|3−δdv .
Z
R3v
µ(v1)hv2i1+γ−δ Z
{|h|≤1}
g(v+h) −g(v) 2 dh
|h|3−δdv .
Z
R3 v
Z
{|h|≤1}
µ(v1−h)+µ(v1)
hv2i1+γ−δg(v)2 dh
|h|3−δdv,
where we have used the change of variables v+h → v for the factor g(v+h). As in the proof of Lemma 2.8, by assuming v=(0,0,|v|), we introduce the polar coordinates
h=(ρsinϑcosφ, ρsinϑsinφ, ρcosϑ), ϑ∈[0, π], φ∈[0,2π], ρ >0.
Since|v1|=|v||cosϑ|,|v1−h|=| |v|cosϑ−ρ|and|v2|=|v|sinϑ, by using the change of varible|v|cosϑ=r, we obtain
A1. Z
R3 v
g(v)2 Z 1
0
1 ρ1−δ
× Z |v|
−|v|
(1+|v|2−r2)(1+γ−δ)/2
|v|
e−|r−ρ|2/2+e−r2/2 dr
! dρdv.
Similarly, if 1+2s−δ >1, then we have A2≤
Z
R3v
Z
{|h|≥1}
Z
y∈Eh
|y|1+γ+2s−δ
|h|1+2s−δ µ(v+y) g(v+h) −g(v)2
dvdhdy
|h|2 .
Z
R3 v
g(v)2 Z ∞
1
1 ρ1+2s−δ
× Z |v|
−|v|
(1+|v|2−r2)(1+γ+2s−δ)/2
|v|
e−|r−ρ|2/2+e−r2/2 dr
! dρdv. If 1+γ+2s−δ≥0, then
K(v, ρ)= Z |v|
−|v|
(1+|v|2−r2)(1+γ+2s−δ)/2
|v|
e−|r−ρ|2/2+e−r2/2 dr
≤ hvi(γ+2s−δ)/2 Z |v|
−|v|
e−|r−ρ|2/2+e−r2/2
dr.hviγ+2s, which shows
A2. Z
R3 v
g(v)2 Z ∞
1
K(v, ρ) ρ1+2s−δdρdv.
Z
hviγ+2sg(v)2dv. (2.7)
On the other hand, if 1+γ+2s−δ <0 and|v| ≥3, then K(v, ρ).
Z |v|
0
(|v|2−r2)(1+γ+2s−δ)/2
|v|
e−|r−ρ|2/2+3e−r2/2 dr .|v|(−1+γ+2s−δ)/2
Z |v|
0
|v| −r(1+γ+2s−δ)/2
e−|r−ρ|2/2+3e−r2/2 dr .hviγ+2s+|v|(−1+γ+2s−δ)/2
Z |v|
|v|/2
|v| −r(1+γ+2s−δ)/2
3e−|r−ρ|2/2dr, because
Z |v|
0
|v| −r(1+γ+2s−δ)/2
e−|r|2/2dr.|v|(1+γ+2s−δ)/2 Z |v|/2
0
e−|r|2/2dr +e−|v|2/8
Z |v|
|v|/2
|v| −r(1+γ+2s−δ)/2
dr,
where we have used that (1+γ+2s−δ)/2 > −1 for smallδ > 0 that follows from the assumptionγ >−3. We now consider
Z ∞
1
dρ ρ1+2s−δ
Z |v|
|v|/2
|v| −r(1+γ+2s−δ)/2
e−|r−ρ|2/2dr
≤ Z |v|
|v|/2
|v| −r(1+γ+2s−δ)/2 Z
{|r−ρ|≤√
2 log|v|}
(|v|/3)−(1+2s−δ)dρ dr
+ Z |v|
|v|/2
|v| −r(1+γ+2s−δ)/2 Z
{|r−ρ|≥√
2 log|v|}
|v|−1dρ ρ1+2s−δ
dr
.
|v|(1+γ+2s−δ)/2p
2 log|v|+|v|(1+γ+2s−δ)/2
.hvi(1+γ+2s)/2.
Therefore, in the case when 1+γ+2s−δ <0, we also have (2.7). Similarly, we have
